The general strategy for the overlaps will mirror the one-particle case. First we introduce some definitions and auxiliary functions.
We call a set of Bethe rapidities λN zero-free, if there is no subset ofλN where the sum of the rapidities is zero. Accordingly, the set lN is zero free, when there is no subset of the l-variables such that their product is 1. States with the pair structure are clearly not zero-free: they are the exceptional states that lead to non-zero overlaps.
Here we investigate overlaps with more general integrable initial states. For simplicity we still restrict ourselves to product states, but we allow for an arbitrary two-site state, thus we consider
|Ψi=⊗L/2j=1|ψi, |ψi ∈ Hj ⊗ Hj+1. (4.16) In the XXZ chain all two-site states are integrable [10], but in models with higher dimension-al locdimension-al spaces the integrability condition puts a restriction on |ψi. Note that the one-site invariant product state considered above is a special case of such two-site states.
The overlap with the reference state is
hΨ|Ωi= (ψ00)L/2, (4.17)
whereψ00 denotes the two-site overlap between the initial state and the reference state. In the compact cases it is given byψ00 =hψ| ↑↑i, and in the non-compact case byψ00=hψ|00i.
For simplicity we focus on cases where ψ00 6= 0. Furthermore we set the normalization to ψ00 = 1, such that the overlap with the reference state is always 1. Initial states with ψ00= 0 can be treated with a limiting procedure, see for example the case of the N´eel state below.
We consider the overlaps
SN(λN) = hΨ|λNi (4.18)
with the Bethe states given in (2.30). It follows from the explicit form of the wave function that every such an overlap is a rational function of the l-variables. The L dependence is hidden in the summation limits. We will show below that for zero-free sets the summations can be performed explicitly, yielding formulae that only involve the lj and aj = (lj)L for eachj, but they do not depend on the volume L in any other way.
Let us therefore introduce the functionSN(λN,aN), which is obtained after these formal manipulations, and after introducing thea-variables:
SN(lN,aN) = hΨ|λNisummed. (4.19) Regarded as a function of a total number of 2N variables, this function does not depend on L anymore. It follows from the form of the wave function and the real space summations that these functions can always be written as
SN(lN,aN) = X
σ∈SN
Y
j>k
f(lσj, lσk)BN(σlN, σaN), (4.20)
where BN is the “kinematical” part of the overlap, which arises from a simple real space summation. It depends on the initial state; explicit formulae will be given below. In the formula above it is understood that σlN, σaN are the permutations of the corresponding ordered sets, namely
σlN ={lσ1, lσ2,· · · , lσN}, σaN ={aσ1, aσ2,· · · , aσN}. (4.21) The quantity BN for some special cases was already defined and computed in [4]. An analogous computation for a non-integrable overlap was performed recently in [35].
Let us also define the function ˜SN(lN) which is obtained from SN by the formal sub-stitution of the Bethe equations. This means that for each aj we substitute the r.h.s. of the corresponding equation from (2.31). It is clear from the above that ˜SN is a symmetric rational function of the setlN.
Theorem 1. The rational function S˜N(lN) is identically zero.
Proof. The function ˜SN does not depend on the volume anymore, it only depends on the l-variables. In the definition of SN we assumed that the set of rapidities is zero-free. The zero-free sets can not satisfy the integrability condition, therefore their overlaps have to be zero. This implies, that the function ˜SN vanishes for all those sets lN that are zero-free solutions to the Bethe equations forany volume. This means that the rational function ˜SN
vanishes at an infinite number of points, therefore it is identically zero.
The non-vanishing overlaps are obtained from SN by a limiting procedure similar to the two-particle case detailed above. The key observation is that for each pair of rapidities (or l-variables lj,lk) there is an apparent simple pole of SN, which is proportional to
ajak−1
ljlk−1 . (4.22)
In the physical cases, when the a-variables are actually given by aj = (lj)L, such a factor simply produces L. However, it is important that we can substitute the Bethe equation onlyafterthese pole contributions are correctly evaluated. Furthermore, all non-zero terms in the overlap can only come from such terms, because if we substitute the Bethe equations before the limit, we get zero identically.
Now we computeSN for paired rapidities. We regardlN andaN as independent variables in the intermediate steps of the computation. We can still assume that there is a well-defined functiona(l) connecting the l- anda-variables, but we do not require the relationa(l) =lL anymore. We will see below that a recursive computation of the overlaps will require to treat more general a(l) functions.
We will consider the limit
l2j−1l2j →1, a2j−1a2j →1, j = 1, . . . , N/2. (4.23) Let us now investigate the apparent pole at say l1l2 = 1.
Proposition 1. The formal pole of SN around the point l1l2 = 1 is of the form SN(L)∼ a1a2−1
l1l2−1F(λ1)
N
Y
j=3
f(λ1 −λj)f(−λ1−λj)SNmod−2(1, 2, L), (4.24) where SN−2mod is the formal overlap for N −2 particles not including 1 and 2, evaluated with the following modified a-variables:
amodj = f(lj, l1) f(l1, lj)
f(lj,1/l1)
f(1/l1, lj)aj. (4.25) In (4.24) F(λ) is a rational function which carries the dependence on the initial state.
At present we do not have a general proof of this statement. However, we are able to rigorously prove it in concrete cases. This leads to the determination of the functionF(λ).
Examples for this will be shown in the next section.
Eq. (4.24) can be considered as a recursion relation for the overlaps. It is rather similar to the recursion relations for scalar products of Bethe states [17] or form factors [36,37] (see also [38,39]). In fact, the modification rule above is a rather straightforward generalization of a similar rule for scalar products, first derived by Korepin in [17]. However, the origin of the poles is different: in the previous cases in the literature the singularities are the so-called kinematical poles of the scalar products or form factors, which appear when two rapidities in the bra and ket vectors approach each other. On the other hand, here the two rapidities responsible for the pole are within the same Bethe vector, and the apparent singularity is associated with the pair structure. The role of such apparent poles was first recognized in [4], and has been used in [35] to study the largeL behaviour of the overlaps.
It is important that if the original l- and a-variables satisfy the Bethe equations, then the restricted set ofl-variables is still on-shell with respect to the modifieda-variables.
We now investigate the limit of the paired rapidities on the basis of the above recursion relation. Let us therefore introduce the set of “positive” rapiditiesλ+N/2, such that the paired limit is taken as
λ2j−1 →λ+j, λ2j → −λ+j , j = 1. . . N. (4.26) Similar notations are understood for thel- and a-variables.
For future use we introduce one more set of variables which will play an important role.
For each j = 1. . . N/2 we define
mj =m(λj) = −i d
dλlog(a(λ)) λ=λj
. (4.27)
In the original physical caseaj =lLj =eip(λj)L we have mj =p0(λj)L, but generally we will treat the m-variables as independent.
Let us define the function D(λ+N/2,m+N/2) as the limit of the function SN described by (4.26). This is a symmetric function under a simultaneous permutation of its variables. It is a rational function ofλ+N/2 and it is at most linear in each of them-variables. The latter property follows from the fact thatSN has only single poles associated to each pair.
Theorem 2. The function D satisfies the recursion
∂D(λ+N/2,m+N/2|L)
∂m+1 = F(λ+1) p0(λ+1)
N/2
Y
l=2
f(λ¯ +1, λ+l )×D(λ+N/2−1,m+,modN/2−1|L), (4.28)
where we defined the modification rule for the m-parameters
mmod(λ) = m(λ) +ϕ+(λ, λ+1). (4.29) Proof. This follows immediately from (4.24), using also Theorem 1. The modification rule for the m-variables follows from
mmod(λ) =−i d
dλlog(amod(λ)), (4.30)
and using (4.25) we get (4.29).
Theorem 3. The solution of the recursion (4.28) is D(λ+N/2,m+N/2|L) =
N/2
Y
j=1
F(λ+j) p0(λ+j )
Y
1≤j<k≤N/2
f(λ¯ +j , λ+k)×detG+N/2. (4.31)
Proof. Our proof follows the method of Korepin derived originally for the Gaudin determi-nant describing the norm of the Bethe states [17].
First we define a function ˜D(λ+N/2,m+N/2) through
D(λ+N/2,m+N/2|L) =
N/2
Y
j=1
F(λ+j) p0(λ+j )
Y
1≤j<k≤N/2
f(λ¯ +j , λ+k) ˜D(λ+N/2,m+N/2|L). (4.32)
It follows from (4.28) that the linear parts in m+j is given by
∂D(λ˜ +N/2,m+N/2|L)
∂m+j = ˜D(λ+N/2−1,m+,modN/2−1|L), (4.33) where it is understood that m+j is not included in the arguments on the r.h.s. and the modification rule is given by (4.29).
The function ˜D satisfies the following properties:
• It is symmetric in all its variables.
• It is at most linear in each mj.
• It is zero if all mj = 0.
• The linear piece in each mj is given by (4.33).
It is easy to see that the unique solution for this linear recursion with the given properties
The normalized squared overlap is obtained after dividing by the norm (3.60). Using the factorization (3.57) we eventually obtain
The single particle overlap function is thus determined by the function F(λ) which determines the apparent singularity of the off-shell overlap:
u(λ) = |F(λ+j )|2
f(2λ+j )f(−2λ+j). (4.36) With this we have finished outlining our general strategy. What remains to be proven is the fundamental singularity relation (4.24), together with finding the function F(λ) in specific cases. This is presented in the next section.